An information-theoretic analysis of resampling in Sequential Monte Carlo

Abstract

Sequential Monte Carlo (SMC) methods form a popular class of Bayesian inference algorithms. While originally applied primarily to state-space models, SMC is increasingly being used as a general-purpose Bayesian inference tool. Traditional analyses of SMC algorithms focus on their usage for approximating expectations with respect to the posterior of a Bayesian model. However, these algorithms can also be used to obtain approximate samples from the posterior distribution of interest. We investigate the asymptotic and non-asymptotic properties of SMC from this sampling viewpoint. Let P be a distribution of interest, such as a Bayesian posterior, and let P be a random estimator of P generated by an SMC algorithm. We study ... i.e., the law of a sample drawn from P, as the number of particles tends to infinity. We give convergence rates of the Kullback-Leibler divergence KL ... as well as necessary and sufficient conditions for the resampled version of P to asymptotically dominate the non-resampled version from this KL divergence perspective. Versions of these results are given for both the full joint and the filtering settings. In the filtering case we also provide time-uniform bounds under a natural mixing condition. Our results open up the possibility of extending recent analyses of adaptive SMC algorithms for expectation approximation to the sampling setting.